Monte Carlo theory of optical dephasing in LaF3:Pr3+

Recent optical free-induction-decay (FID) measurements of the impurity ion ${\mathrm{Pr}}^{3+}$ in La${\mathrm{F}}_3$ at 2 °K reveal optical homogeneous linewidths of only a few kilohertz, considerably narrower than the inhomogeneous broadening due to crystalline strains (5 GHz) or the static local magnetic fields of the ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ nuclei (100 kHz). In this regime, the homogeneous broadening arises from local field fluctuations, as in NMR, and is due to the ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ nuclei which undergo resonant flip-flops and modulate the ${\mathrm{Pr}}^{3+}$ optical transition frequency. We treat the optical response of a two-level quantum system to an intense coherent field and to fluctuating perturbations using a Monte Carlo computer routine that assumes (1) the La${\mathrm{F}}_3$ crystal structure and (2) a sudden fluorine spin-flip model. This procedure avoids many of the approximations of previous analytic theories of spectral diffusion in magnetic resonance and extends the calculation specifically to optical FID. The decay behavior is obtained by sampling statistically the ${\mathrm{Pr}}^{3+}$ phase history as subgroups of ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ spins flip randomly in space and time. These fluctuations modify the Bloch equations where the solutions for the preparative and post-preparative periods are obtained by numerical integration. In spite of the large lattice size assumed (2250 fluorines), only a few ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ spins contribute substantially to the homogeneous width, a result which shows for the first time that spin-flip correlations are not significant. Furthermore, a ${\mathrm{Pr}}^{3+}$ ion polarizes and detunes the nearest fluorines forming a frozen core that is incapable of resonant spin flipping with the bulk fluorines. We demonstrate that the core grows radially as the ${}_{}{}^{141}{}_{}{}^{}\mathrm{Pr}(I=\frac{5}{2})$ magnetic moment increases with $I_z$, but the Pr optical linewidth changes little, producing essentially one rather than three linewidths. Our calculations utilize no free parameters and predict a Lorentzian line shape of 8.4 kHz half-width at half maximum which compares to the optical FID observation of a 10.1-kHz Lorentzian. The Monte Carlo algorithm is verified further by the static local magnetic broadening of a Pr quadrupole transition which is found to be Gaussian, 82 kHz full width at half maximum, in agreement with a second-moment calculation and current observations.